Final answer:
The square-based prism that minimizes the manufacturing cost with a constant volume of 250 cm^3 is the one with dimensions of 2.5 cm for the side length of the square base and 20 cm for the height, as this configuration results in the smallest surface area.
Step-by-step explanation:
The question asks for the dimensions of a square-based prism (a safety deposit box) that minimize the manufacturing cost, given that its volume is 250 cm3. Considering four different scenarios for the dimensions of the box, we must calculate which of them leads to the least surface area since the cost is often related to the amount of material needed which, in turn, is related to the surface area.
To obtain the surface area for a square-based prism, we use the formula SA = 2s2 + 4sh, where s is the side of the square base, and h is the height of the prism. The volume of the prism is given by V = s2h. Meanwhile, the volume remains constant at 250 cm3. Thus each option provided gives a different surface area:
Option a) SA = 2(5 cm)2 + 4(5 cm)(10 cm) = 50 cm2 + 200 cm2 = 250 cm2.
Option b) SA = 2(10 cm)2 + 4(10 cm)(5 cm) = 200 cm2 + 200 cm2 = 400 cm2.
Option c) SA = 2(20 cm)2 + 4(20 cm)(2.5 cm) = 800 cm2 + 200 cm2 = 1000 cm2.
Option d) SA = 2(2.5 cm)2 + 4(2.5 cm)(20 cm) = 12.5 cm2 + 200 cm2 = 212.5 cm2.
From these calculations, we can see that option d) has the smallest surface area and thus would likely minimize the cost of manufacturing the deposit box.